FTLE and LCS Pranav Mantini

Contents Introduction Visualization Lagrangian Coherent Structures Finite-Time Lyapunov Exponent Fields Example Future Plan

Time-Varying Vector Fields Vector Field defines a vector(v(x)) at every point x on the grid In time variant vector field the vector defined at the points on the grid change with time(v(x,t)). Creating complex patterns and requires sophisticated techniques for analysis and visualization

Applications Thorough analysis of flows plays an important role in many different processes, o Airplane o Car design o Environmental research o And medicine Deepwater Horizon Oil Spill

Mathematical Framework

Visualization Traditionally visualized using Vector Field Topology. Gives a simplified representation of a vector field a dynamical system, with respect to the regions of di ff erent behavior. VFT deals with the detection, classification and global analysis of critical points VFT are significantly helpful for visualizing the time independent vector fields.

Visualization In time varying vector fields, pathline diverge from stream lines and the critical points move. Forces to visualize only at a single point of time. Coherent structures provide a more meaningful representation

Coherent Structures Lagrangian Coherent Structures LCS has gained attention in visualizing time dependent vector fields A set of LCS can represents regions that exhibits similar behavior example, a recirculation region can be delimited from the overall flow and can represent an isolated LCS LCS boundaries can be obtained by computing height ridges of the finite-time Lyapunov exponent fields

Real World Correspondence Confluences Glaciers LCS = InterfacesLCS = Moraines from:

Finite-Time Lyapunov Exponent Fields Scalar Value Quantifies the amount of stretching between two particles flowing for a given time High FTLE values correspond to particles that diverge faster than other particles in the flow field High FTLE Values

Finite-Time Lyapunov Exponent Fields

Lagrangian Coherent Structures Ridge lines in these fields correspond to LCS Height ridges are locations where a scalar field has a local extremum in at least one direction Ridge criterion can be formulated using the gradient and the Hessian of the scalar field Eigenvectors belonging to the largest eigenvalues of the Hessian point along the ridge, and the smallest point orthogonal to the ridge.

Example Velocity Field Time-dependent double gyre Domain [0, 2] x [0, 1]

FTLE and LCS FTLE LCS

Crowd Flow Segmentation & Stability Analysis (CVPR), 2007

Future Plan It is obvious that the LCS are influenced by the 3D geometry. It might be interesting to see how the change in geometry influences the LCS Build Vector Field, Find LCS Change Geometry Estimate LCS

Future Plan Week – 1&2: Estimate LCS for an example Vector Field Week – 3: Build a Vector Field from real world scenario Week – 4: Estimate LCS Week – 5….: Try to estimate LCS based on geometry and other information, in the absence of a vector field

References “Visualizing Lagrangian Coherent Structures and Comparison to Vector Field Topology” Filip Sadlo and Ronald Peikert Computer Graphics Laboratory, Computer Science Department Efficient Computation and Visualization of Coherent Structures in Fluid Flow Application. Christoph Garth, Florian Gerhardt, Xavier Tricoche, Hans Hagens

Any Questions